Problem: "How are we ever going to build this bridge?" asks Omkar looking out across the raging river. "Let's start by finding the distance to the big rock on the other side." Melissa replies. Moving $100$ meters along the river, Melissa looks back and measures the angle between Omkar and the big rock: $33^\circ$. Melissa then instructs Omkar to measure the angle between Melissa and the big rock. From his vantage point, Omkar sees an angle of $98^\circ$ between Melissa and the big rock. What is the distance across the river from Omkar to the big rock? Do not round during your calculations. Round your final answer to the nearest meter.
Explanation: Converting the problem into geometrical terms Our problem can be modeled by the following triangle $\triangle ABC$, where we want to find $AC=d$. Because the interior angles of a triangle must add to $180^\circ$, we know that $\angle C=49^\circ$. $A$ $B$ $C$ $100\text{ m}$ $33^\circ$ $98^\circ$ $49^\circ$ $d$ Since we are given one side length and all angle measures, we can use the law of sines. Using the law of sines $\begin{aligned} \dfrac{\sin(C)}{AB}&=\dfrac{\sin(B)}{AC}\\\\ \dfrac{\sin(49^\circ)}{100} &= \dfrac{\sin(33^\circ)}{d} \gray{\text{Substitute}} \\\\ d \cdot \sin(49^\circ) &= 100 \cdot \sin(33^\circ) \\\\ d &= \dfrac{100 \cdot \sin(33^\circ) }{\sin(49^\circ) } \\\\ d &\approx 72 \,\text{m} \end{aligned}$ The answer The distance across the river from Omkar to the Big Rock is $72$ meters.